linear-algebra


Linear iterative solver vs direct solver stability


Is iterative solver more stable than direct solver based on LU factorization. For LU based solver, we always have cond(A) < cond(L) * cond(U), so factorization amplifies numerical inaccuracy. So in the event of an ill conditioned matrix A, whose condition number is large than 1e10, will it be better off using iterative solver for stability and numerical accuracy?
There are two factors involved into answering your question.
1) The physical system you are analyzing is ill-conditioned by itself (in mechanical terms, the system is pretty "loose", so its equilibrium state may vary greatly depending on just a small variation in the boundary conditions)
2) The physical system is OK, but the matrix has not been scaled properly before the solution process begins.
In the first case, there isn't much you can do: the physical system is inherently unstable. Consider applying different boundary conditions, for example.
In the second case, a preconditioner should be helpful; for example, the Jacobi preconditioner makes the matrix having all diagonal values equal to 1. In this case, the iterations are more likely to converge.The condition ratio of 1e10 shouldn't represent too much trouble, provided a preconditioning is used.

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